Connections Between Voting Theory and Graph Theory

Transcription

1 Connections Between Voting Theory and Graph Theory Debbie Berg Francis Su, Advisor Michael Orrison, Reader December, 2005 Department of Mathematics

2 Copyright c 2005 Debbie Berg. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copying is by permission of the author. To disseminate otherwise or to republish requires written permission from the author.

3 Contents Acknowledgments ix 1 Introduction Definitions Background Motivation Graphs Representations with Graphs Properties of Agreement Graphs Set Intersections, Perfect Graphs, and Voting in Agreeable Societies Introduction Definitions Helly s Theorem and Super-Agreeable Societies Graph Representations Interval Graphs and Perfect Graphs (k, m)-agreeable Societies Speculation and Open Questions Convex Sets with k of Every m Meeting Introduction Definitions Helly s Theorem Relationships Between Agreement and Piercing Numbers Piercing Numbers and Agreement Graphs Families of Societies Summary Cubes

4 iv Contents 4.9 Speculation and Open Questions The Prison Warden s Dilemma: A Characterization of 3-Convexity 37 Abstract Discussion Conclusions 47 Bibliography 51

5 List of Figures 1.1 One Method of Labeling the Unit Interval Linear Graph of a (2, 3, 6)-Society Agreement Set of a Society with Agreement Number Four Agreement Graph of the (2, 3, 6)-Society from Figure Linear Graph of a (2, 2, 6)-Society Agreement Graph of a (2, 3, 6)-Society with Two Components Agreement Graph of a Similar Society with Three Components Labeling the Unit Interval for Bipartisan Politics Linear Graph of a (2, 3, 6)-Society Agreement Set of a Society with Agreement Number Four Linear Graph of a (2, 2, 6)-Society Agreement Graph of a (2, 3, 6)-Society Agreement Graph of a (2, 3, 6)-Society with Two Components Agreement Graph of a Similar Society with Three Components Two Graphs Pasted Along a Common Subgraph Linear Graph of a (2, 3, 6)-Society Agreement Set of a Society with Agreement Number Four Collection of Sets with Piercing Number Two A (2, 3, 10)-Society of Convex Sets A Collection of Ten Sets with Piercing Number Five Agreement Graph of Figure (2, 2)-Society with Agreement Number Two A (3, 7)-Society with Agreement Number Three and Piercing Number Five A (3, 7)-Society with Three Pierced Sets Agreement Graph of Figure 4.8 with 3 Intersections Shaded Edges of a 4-Dimensional Cube Pierced with Eight Points.. 35

6 vi List of Figures 5.1 Union of a Star around p and a Convex Set A Non-Closed Counter Example to Theorem Construction Used in the First Proof of Lemma Construction Used in the Alternate Proof of Lemma Star-Shaped and Convex Sets in a Five-Pointed Star A Regular Seven-Pointed Star Regions in a Regular Seven-Pointed Star A (3, 4, 6)-Society with Piercing Number Three

7 List of Tables 4.1 Edges of d-cubes as Sets in a Collection

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9 Acknowledgments I d like to thank Professor Francis Su for being my thesis advisor, despite the additional hardship from my thesis being a spring-fall thesis. I d also like to thank Professor Michael Orrison, both for being my second reader and for suggesting I start my thesis a semester earlier than is usual. Additionally, I d like to thank Tyler Seacrest for allowing me to discuss my ideas and conjectures with him, which eventually led to our paper on 3- convexity.

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11 Chapter 1 Introduction Mathematical concepts have aided the progression of many different fields of study. Math is not only helpful in science and engineering, but also in the humanities and social sciences. Therefore, it seemed quite natural to apply my preliminary work with set intersections to voting theory, and that application has helped to focus my thesis. Rather than studying set intersections in general, I am attempting to study set intersections and what they mean in a voting situation. This can lead to better ways to model preferences and to predict which campaign platforms will be most popular. Because I feel that allowing people to only vote for one candidate results in a loss of too much information, I consider approval voting, where people can vote for as many platforms as they like. 1.1 Definitions If we are concerned with a single issue where opinions range from one extreme to another, we can represent voters opinions using a linear system. Consider the set [0, 1] R and assign 0 to one extreme view and 1 to the other. Then every view, or platform, in between can also be represented by a real number between 0 and 1. See Figure 1.1 for an example of how we might label the interval in terms of voting preferences. Let the set of platforms for whom person i will vote for be represented by the set A i. For convenience, we can distribute the sets along the vertical axis, as shown in Figure 1.2. For the purpose of creating a realistic model, we require that A i be a closed interval (or the empty set). Thus, if person i will vote for platforms p and q, he will vote for a platform which is intermediate.

12 2 Introduction Figure 1.1: One Method of Labeling the Unit Interval A society, S, is a group of voters together with the platforms they can agree with, which we call X. The order of a society, S, is the number of voters in the society. A (k, n)- society is a society such that for any subset of n voters, there is at least one platform that at least k of them can agree upon. Notice that the order of a (k, n)-society is at least n. To further describe the society, we say a (k, n, t)-society is a (k, n)-society with order t. Figure 1.2: Linear Graph of a (2, 3, 6)-Society The agreement number of a platform, a(p), is the number of voters who approve of platform p. The agreement number a(s) of a society S is the maximum agreement number over all platforms in X, so a(s) = max p X a(p). The agreement proportion of S is simply the agreement number of S divided by the order of S, or a(s)/ S. This concept is useful when we are interested in percentages of the population rather than the number of voters. The agreement set of S consists of platforms that receive a(s) votes. Notice that Figure 1.3 shows a society with agreement number 4, and the shaded rectangles cover the agreement set. It is again important to remember that the vertical axis only makes the graph easier for us to interpret and has no mathematical meaning. Finally, we say that a society is super-agreeable if a(s) = S. The distance between voters, d(a, b), is the minimum number of other voters v 1, v 2,..., v d(a,b) required such that A a A v1 = {}, A v1 A v2 =

13 Background 3 Figure 1.3: Agreement Set of a Society with Agreement Number Four {},..., A vd (a,b) A b = {}. In Figure 1.3, we see that the distance between voters 2 and 3, for example, is 1, because there are no platforms that both 2 and 3 will vote for, but 2 and 3 both have platforms in common with voter 1. If no set of voters will satisfy this condition, we say d(a, b) =. If d(a, b) = 0, then we say the voters are adjacent. In Figure 1.3, for instance, we see that voters 1 and 2 are adjacent. Upon closer inspection, we realize that voter 1 is actually adjacent to every other voter. 1.2 Background We realize that a society is simply a set of voters, so work concerning set intersections can be applied to this problem. The most well known theorem in this area is Helly s theorem [1], proven in Helly s theorem states that given t convex sets in R d, if every d + 1 of them intersect at a common point, then they all intersect at a common point. The KKM lemma [10], proven in 1929, is similar to Helly s theorem, but is concerned with set intersections on simplices. A more recent theorem, proven by Neidermaier and Su in a paper still in progress, generalizes Helly s theorem to non-convex sets on trees. These theorems gave me a good place to start research, as I could see how they applied to a linear representation of voting theory. 1.3 Motivation The concept of agreeable societies of voters is useful in voting theory for many reasons. For instance, it can be used to determine the minimum num-

14 4 Introduction ber of platforms that are necessary such that everyone has some platform of which he or she approves. It can also lead to comparisons of voters in those societies in an attempt to determine what kind of people approve of certain platforms. Thus, using these modeling techniques, politicians can attempt to satisfy the greatest number of voters possible. Another application of this is in consensus theory. Consensus theory is related to voting theory in that it is used for decision making when opinions may not agree. However, consensus theory is more flexible than voting theory, and it attempts to use more information than is generally available in voting theory. Consensus theory tends to be used in a managerial setting. Although it encompasses more than voting theory does, the results of this thesis may be quite useful to consensus theory in general. Additionally, since this thesis stemmed from work in a variety of other areas, it seems certain that conclusions formed in the context of voting theory will be applicable to many other branches of mathematics. In Chapter 2, we consider methods of using graphs to explore the properties of different societies, and in Chapter 3, we examine this idea more thoroughly. Chapter 4 introduces concepts related to the piercing number of a graph and explains how the piercing number and agreement number are related. Chapter 5 considers a possible generalization of convexity. In this chapter, we find a necessary and sufficient condition for a set to be 3- convex, but unfortunately, 3-convex sets do not model societies in a realistic way. We draw conclusions, both mathematical and otherwise, in Chapter 6.

15 Chapter 2 Graphs 2.1 Representations with Graphs We now examine methods of pictorially representing the agreeableness of a society. Consider a (k, n, t)-society, and let each vertex in an agreement graph G represent a voter. Draw an edge between two voters if the voters are adjacent. We notice that a super-agreeable society will produce a complete graph. Additionally, if d(a, b) =, then there is no path from a to b, so we say that a and b are in different components of the agreement graph. Figure 2.1: Agreement Graph of the (2, 3, 6)-Society from Figure 1.3 Theorem 2.1 (Helly). A (2, 2, t)-society must contain at least one platform that all t voters will vote for. This follows as a consequence of Helly s theorem for dimension 1. It

16 6 Graphs also results from a theorem Niedermaier and Su proved for set coverings of trees in a paper still in progress. Below I will give an alternate proof from a different viewpoint. Proof: Since each voter agrees on at least one platform with every other voter, we see that the sets A i must be non-empty. Thus, each A i is a nonempty closed interval in [0, 1]. We notice that n = min i {max{p A i }} max i {min{p A i }} = m, because all A i are adjacent and must share at least one platform p. Therefore, all intervals contain the platforms in the non-empty interval [m, n], so there is at least one platform that all voters will vote for. Figure 2.2: Linear Graph of a (2, 2, 6)-Society 2.2 Properties of Agreement Graphs Theorem 2.2. The agreement graph of a (k, n)-society has no more than n k + 1 disjoint components. Proof: Consider a graph G with at least n k + 2 disjoint components, and let S be the set of vertices such that we choose one vertex from each component and k 2 other arbitrary vertices. Then there are no more than k 1 vertices of S in a single component of G. Since voters in distinct components will not agree to vote for any particular platform, there is no set of k voters who will all vote for the same platform, so no choice of n voters is such that k of those voters will agree on a platform. Therefore, the agreement graph of a (k, n)-society has no more than n k + 1 disjoint components. An example may help make this reasoning even more clear. We see that Figure 2.3 has 2 components. I claim that this is a (2, 3, 6)-society. We

17 Properties of Agreement Graphs 7 can assume the contrary and attempt to find three vertices such that no pair is adjacent. First, choose an arbitrary vertex v, which must be in some triangle. Since triangles are complete, any other vertex in that triangle is adjacent to v, so we must choose a vertex w in the other triangle. However, we have now chosen vertices from each triangle, so we cannot find a third vertex such that it is not adjacent to either of the first two. Thus, this is a (2, 3, 6)-society, and we see that it has n k + 1 = 2 components. We can then consider this (2, 3, 6)-society with an additional vertex in a new component, as shown in Figure 2.4. Is this a (2, 3, 7)-society? We see that it is not, because the three circled vertices form a set of three vertices such that no two are adjacent. The addition of a third component is what made the difference, since all sets of three independent vertices involve the new component. Thus, we see that the agreement graph cannot have n k + 2 = 3 components. This reasoning generalizes to any choice of k, n, and t such that k n t. Figure 2.3: Agreement Graph of a (2, 3, 6)-Society with Two Components Figure 2.4: Agreement Graph of a Similar Society with Three Components Corollary 2.1. The agreement number of a (k, n, t)-society is at least t/(n k + 1). In particular, the agreement number of a (2, 3, t)-society is at least t/2.

18 8 Graphs Proof: By Theorem 2.2, we know the agreement graph of a (k, n, t)- society has no more than n k + 1 disjoint components. Therefore, since every voter must be in some component, by the pigeonhole principle, there must be some component with at least t/(n k + 1) voters. In particular, the agreement proportion of a (2, 3, t)-society is at least 1 2, so there must be at least one platform that at least half of the population will vote for. These results are extremely reminiscent of the piercing number of a set of sets [7]. Although they were developed independently, the piercing number and the agreement number are related. The piercing number of a society of sets is the minimum number of points required such that every set contains at least one point. We notice that the agreement number of a society is equivalent to the maximum number of sets that can be taken care of with a single point. Thus, if there are n sets in a society and the agreement number of the society is k, we know that the piercing number is less than or equal to n k + 1.

19 Chapter 3 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies I spent the summer of 2005 at Harvey Mudd College, doing research with Professor Su. We co-authored a paper with Robin Thomas, Serguei Norine, and Paul Wollan that pertains to this thesis. The paper is currently being prepared for submission for publication, and it is included here in its complete form. 3.1 Introduction When is agreement possible? An important aspect of group decision-making is the question of how a group makes a choice when individual preferences may differ. Clearly, people cannot all have their ideal preferences, i.e, the options that they most desire, if those ideal preferences are different. However, for the sake of agreement, people may be willing to accept as a group choice an option that is merely close to their ideal preferences. A good example of such a situation is voting for candidates along a political spectrum. We normally think of this spectrum as one-dimensional, with conservative positions on the right and liberal positions on the left, as in Figure 3.1. While we may represent our ideal preference at some point x on this interval, we might be willing to vote for a candidate that positions himself at some point close to x. In this article, we ask the following: given such preference sets on a

20 10 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies Figure 3.1: Labeling the Unit Interval for Bipartisan Politics political spectrum, when can we guarantee the existence of some fraction (say, a majority) of the population who would agree on some candidate? By agree, we mean in the sense of approval voting, in which voters declare which candidates they find acceptable. Approval voting has not yet been adopted for political elections in the United States. However, many scientific and mathematical societies, such as the Mathematical Association of America and the American Mathematical Society, use approval voting for their elections. Additionally, countries other than the United States have used approval voting or an equivalent system. For details, see Brams and Fishburn [4], who give many reasons why they believe approval voting is advantageous. In what follows, our study of agreeability will help us understand when we can guarantee a majority under approval voting. Suppose that approval voting were used in the 2003 California gubernatorial recall election, with 135 candidates in the mix [5]. We might imagine these candidates positioned at 135 points on the line. If each California voter approved of candidates within some range of positions (call this the voter s approval set), we might wonder if and when there might be a point on the interval covered by a majority of the voter approval sets, i.e., a platform on which a majority of the voters agree. An extremely strong condition that guarantees this (and quite a bit more) is that every pair of voters agrees on some platform, i.e., each pair of approval sets has a non-empty intersection. If this condition is met, call the society of voters super-agreeable. This local condition guarantees a strong global property, namely, that there is a platform of which every voter approves! As we shall see in Theorem 3.3, this is a consequence of Helly s theorem about intersections of convex sets. In this article, we consider a variety of similar theorems. For instance, we relax the condition above and call a society agreeable if among every 3 voters, there is some pair of voters who agree on some platform. Then we prove the following: Theorem 3.1 (The Agreeable Society Theorem). In an agreeable society, there is a platform which has the approval of a majority of voters, i.e., a winning platform.

21 Definitions 11 For example, Figure 3.2 shows approval sets for an agreeable society of six voters, and indeed there are platforms of which a majority of voters approve. As another application of our theorem, consider a situation in which each voter s approval set spans 1/3 of the total interval. Then the theorem above guarantees a winning platform. We also consider other degrees of agreeability and prove more general results in Theorems 3.5 and 3.9. As we shall see, these questions motivate the study of theorems about set intersections and perfect graphs, since they have natural interpretations in this voting context. 3.2 Definitions In this section, we fix some terminology and explain some of the basic concepts upon which our results rely. Let us suppose that the set of possible preferences is modeled by a set X, called the spectrum. Each element of the spectrum is a platform. Assume that there is a finite set of voters, and each voter v has an approval set A v of platforms. The set X together with all the approval sets of all voters will be called a society. Of particular interest to us will be the case where X = [0, 1]. The political spectrum is often modeled this way, but this is by no means limited to politics. Another situation that is well-described by this model is preferences over a single issue where opinions range from one extreme to another with 0 as one extreme view and 1 as the other. See Figure 3.1 for an example of how we might label the interval in terms of voting preferences. In Figure 3.2, for ease of display, we have separated the approval sets vertically so that they can be distinguished. Figure 3.2: Linear Graph of a (2, 3, 6)-Society A society is (k, m)-agreeable, where 1 k m are integers, if it has at least m voters, and for any subset of m voters, there is at least one platform

22 12 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies that at least k of them can agree upon. For a society S, the agreement number of a platform, a(p), is the number of voters in S who approve of platform p. The agreement number a(s) of a society S is the maximum agreement number over all platforms in the spectrum, so a(s) = max p X a(p). Note that a society is super-agreeable if a(s) = S. The agreement proportion of S is simply the agreement number of S divided by the order of S, or a(s)/ S. This concept is useful when we are interested in percentages of the population rather than the number of voters. The agreement set of S consists of platforms that receive a(s) votes, and this is a subset of X, the set of all possible platforms of which voters can approve. Figure 3.3 shows a society with agreement number 4, and the shaded rectangles cover the agreement set. The agreement number of a platform, a(p), is the number of voters who approve of platform p. Figure 3.3: Agreement Set of a Society with Agreement Number Four 3.3 Helly s Theorem and Super-Agreeable Societies Let us say that a society is R d -convex if the spectrum is R d and each approval set is a closed convex subset of R d. Note that a linear society is an R 1 -convex society. In case of R d -convex societies, work concerning set intersections can be applied to the agreement number problem. The most well known theorem in this area is Helly s theorem. This theorem was proven by Helly in 1913, but the result was not published until 1921, by Radon [14].

23 Helly s Theorem and Super-Agreeable Societies 13 Theorem 3.2 (Helly). Given t convex sets in R d where d < t, if every d + 1 of them intersect at a common point, then they all intersect at a common point. Note that in dimension 1, Helly s condition for approval sets is equivalent to the condition for a super-agreeable society. The conclusion of Helly s theorem therefore has a nice interpretation: Theorem 3.3 (The Super-Agreeable Society Theorem). A (2, 2)-agreeable society must contain at least one platform that is acceptable to all voters. A proof of Helly s theorem for general d may be found in [11], but below we give an alternate proof for the case d = 1, e.g., Theorem 3.3. Proof. Since each voter agrees on at least one platform with every other voter, we see that the sets A i must be non-empty. Thus, each A i is a nonempty closed interval in [0, 1]. Let x = max i {min{p A i }} and y = min j {max{p A j }}. Figure 3.4: Linear Graph of a (2, 2, 6)-Society We claim that x y. Why? Let i be the voter whose approval set minimum is maximal, and let j be the voter whose approval set maximum is minimal. Note that i and j are adjacent, and the only way this could be the case is if x y. Therefore, all approval sets contain the platforms in the non-empty interval [x, y], so there is at least one platform that all voters will vote for. Besides Helly s theorem, another famous theorem about set intersections is the KKM lemma [10], which is concerned with set intersections on simplices. More recently, Niedermaier and Su [12] proved a set intersection theorem on trees that generalizes both Helly s theorem and the KKM

24 14 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies lemma to this context. Since a line is a tree, Theorem 3.3 for d = 1 can also be proved from their results. Here is an example demonstrating that the convexity assumption is essential. Let n 2 be an integer and let the spectrum of a society S consist of all 2-element subsets of {1, 2,..., n}. Let S have n voters numbered 1, 2,..., n, and let the approval set of voter i consist precisely of those 2- element subsets of {1, 2,..., n} that include i. Then S is a (2, 2)-agreeable society with agreement number 2, which is in sharp contrast with Theorem Graph Representations If we are to understand other kinds of agreeability beyond super-agreeability, it will be helpful to examine methods of pictorially representing the agreeableness of a society. A graph G consists of a finite set V(G) of vertices and a set E(G) of 2-element subsets of V(G), called edges. If e = {u, v} is an edge, then we say that u, v are the ends of e, and that u and v are adjacent in G. We use uv as another notation for the edge e. Given a society S, we construct an agreement graph G of S by letting each vertex represent a voter and drawing an edge between two voters if the voters are adjacent, meaning that there is some candidate of which they both approve. Note that a super-agreeable society will produce a complete graph. Also, the distance between two voters (as defined earlier) is just the graph distance in the agreement graph. Moreover, if d(a, b) =, then there is no path from a to b, so we say that a and b are in different components of the agreement graph. We now have some tools with which to study (k, m)-agreeability. Theorem 3.4. The agreement graph of a (k, m, n)-society has no more than m k + 1 disjoint components. Proof. Consider a graph G with at least m k + 2 disjoint components, and let S be the set of vertices such that we choose one vertex from each component and k 2 other arbitrary vertices. Then there are no more than k 1 vertices of S in a single component of G. Since voters in distinct components will not agree to vote for any particular platform, there is no set of k voters who will all vote for the same platform, so no choice of m voters is such that k of those voters will agree on a platform. Therefore, the agreement graph has no more than m k + 1 disjoint components.

25 Graph Representations 15 Figure 3.5: Agreement Graph of a (2, 3, 6)-Society An example may help make this reasoning even more clear. We see that Figure 3.6 has 2 components. We claim that this is a (2, 3, 6)-society. We can assume the contrary and attempt to find three vertices such that no pair is adjacent. First, choose an arbitrary vertex v, which must be in some triangle. Since triangles are complete, any other vertex in that triangle is adjacent to v, so we must choose a vertex w in the other triangle. However, we have now chosen vertices from each triangle, so we cannot find a third vertex such that it is not adjacent to either of the first two. Thus, this is a (2, 3, 6)-society, and we see that it has m k + 1 = 2 components. Figure 3.6: Agreement Graph of a (2, 3, 6)-Society with Two Components We can then consider this (2, 3, 6)-society with an additional vertex in a new component, as shown in Figure 3.7. Is this a (2, 3, 7)-society? We see that it is not, because the three circled vertices form a set of three vertices such that no two are adjacent. The addition of a third component is what made the difference, since all sets of three independent vertices involve the new component. Thus, we see that the agreement graph cannot have

26 16 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies m k + 2 = 3 components. This reasoning generalizes to any choice of m, k, and n such that k m n. Figure 3.7: Agreement Graph of a Similar Society with Three Components Theorem 3.5. The agreement number of a (k, m, n)-society is at least n/(m k + 1). In particular, the agreement number of a (2, 3, n)-society is at least n/2. Proof. By Theorem 3.4, we know the agreement graph of a (k, m, n)-society has no more than m k + 1 disjoint components. Therefore, since every voter must be in some component, by the pigeonhole principle, there must be some component with at least n/(m k + 1) voters. In particular, the agreement proportion of a (2, 3, n)-society is at least 1 2, so there must be at least one platform that at least half of the population will vote for. These results are reminiscent of the piercing number Π of a collection of sets [7]. Although they were developed independently, the piercing number and the agreement number are related. The piercing number of a collection of sets is the minimum number of points required such that every set contains at least one of those points. Note that the agreement number of a society is equivalent to the maximum number of sets that can be pierced by a single point. Thus, if there are n sets in a society S, then Π n a(s) + 1. Additionally, since the piercing number is the smallest number of points such that each set contains at least one point and the agreement number is the largest number of sets that can be pierced by one point, the total number of sets must be at least the agreement number times the piercing number. In other words, Π n/a(s).

27 Interval Graphs and Perfect Graphs Interval Graphs and Perfect Graphs The chromatic number of a graph G, written as χ(g), is the minimum number of colors necessary to color vertices such that no adjacent vertices have the same color. This can tell us information about relationships between voters preference sets. In general, the chromatic number of an agreement graph rises as the agreeability of the society rises. The clique number of G, written ω(g), is the greatest integer n such that K n G. In an agreement graph, the clique number is the agreement number of the society. Notice that in all cases, χ(g) ω(g). A graph G is called an interval graph if every vertex x represents a real interval I x and xy E(G) if and only if I x I y =. We notice that because we restrict voters to approving of contiguous closed intervals (or the empty set), agreement graphs are interval graphs. Given a cycle on n vertices, an edge e is a chord if it is adjacent to two vertices of the cycle but is not in the cycle itself. If a graph G is such that any cycle of length greater than three has a chord, it is a chordal graph, sometimes called a triangulated graph. Theorem 3.6. Interval graphs are chordal. Proof. We prove the contrapositive: if a graph is not chordal, it cannot be an interval graph. Let G be a non-chordal graph, so there is some cycle C of order n > 3 such that C contains no chords. Label the vertices of the cycle v 1, v 2,..., v n such that v 1 is adjacent to v n, which we write as v 1 v n, and v i v i+1 for i {1, 2,..., n 1}. Assume by way of contradiction that G is an interval graph, and let I i = [a i, b i ] be the interval corresponding to vertex v i. Without loss of generality, assume a 1 a i for all i {2, 3,..., n}. Because [a 1, b 1 ] [a 2, b 2 ] =, [a 2, b 2 ] [a 3, b 3 ] =, and [a 1, b 1 ] [a 3, b 3 ] =, we see that a 1 a 2 b 1 < b 2. By symmetry, a i a i+1 b i < b i+1 for all i {1, 2,..., n 2}. We know a n 1 a n, so a 1 < a n. Thus, since I n 1 I n =, a n b n 1 b n. Also, since I 1 I n = but I 1 I n 1 =, we know a n b n 1 a 1, which is a contradiction. Therefore, G is not an interval graph, so all interval graphs must be chordal graphs. Theorems 3.7 and 3.8 and their proofs are adapted from Diestel s Graph Theory [6]. We first introduce some terminology. Note that pasting graphs G 1 and G 2 along a subgraph S (G 1 G 2 ) means that we essentially overlap some of the identical parts of G 1 and G 2. In Figure 3.8, S is composed of edges in blue and their associated vertices, and the two graphs on the

28 18 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies left are pasted along S to produce the graph on the right. Additionally, an induced cycle is simply a cycle with no chords. Note that for S V(G), we write G[S] to mean the graph induced by S, so G[S] contains all vertices in S and all edges in G that have both endpoints in S. Finally, a minimal separating set between vertices x and y consists of the fewest number of vertices that we must remove such that x and y are in different components of the resulting graph. Figure 3.8: Two Graphs Pasted Along a Common Subgraph Theorem 3.7. A graph G is chordal if and only if it can be constructed recursively by pasting two chordal graphs along a complete graph. Proof. Let G be a graph constructed of two chordal graphs, G 1 and G 2, pasted along a complete subgraph S (G 1 G 2 ) and let C be a cycle in G. If C (G G 1 ) and C (G G 2 ) are both non-empty, then the cycle must contain at least two disjoint edges of S. However, S is a complete graph, so the vertices in those two disjoint edges are all adjacent. Thus, C contains a chord. Therefore, any induced cycle of G must be a subset of either G 1 or G 2. By definition, this means any induced cycle of G is an induced cycle of G 1 or G 2, which are chordal graphs. Thus, all induced cycles of G are triangles, so G is chordal. We prove the other direction by induction. Let G be a chordal graph. If G = 1, the construction is trivial. In fact, if G is complete, the construction is trivial, since all induced cycles of a complete graph are triangles. Thus, we assume G is not complete, G > 1, and all chordal graphs smaller than G can be constructed in the manner specified. Because G is not complete, we can find two vertices, x and y, that are not adjacent. Let M V(G) be a minimal separating set between x and y, and consider G M. Let X be the component of G M containing x. Let G 1 = G[M V(X)] and G 2 = G X. Then G can be constructed by pasting G 1 and G 2 along G[M]. Since G 1 and G 2 are both induced subgraphs of G, they must be chordal,

29 Interval Graphs and Perfect Graphs 19 by the inductive hypothesis. Thus, we simply need to show that G[M] is complete. Suppose the contrary, that G[M] is not complete. Then we can find two non-adjacent vertices in G[M], which we ll label a and b. Because G[M] is a minimal separating set, each must be adjacent to some vertex in X. Since X is a single component, there must be some minimal path P between a and b contained entirely in X, which is in G 1. Similarly, there must be some minimal path P between a and b contained entirely in the component containing y, which is in G 2. Thus, P P is a cycle of length at least 4. However, since this is constructed from minimal paths, this cycle has no chords, which contradicts the fact that G is chordal. Thus, we cannot find non-adjacent vertices in G[M], so G[M] is complete. Therefore, we can construct an arbitrary chordal graph G by taking two chordal graphs, G 1 and G 2, and pasting them along a complete graph, G[M]. If every induced subgraph H of a graph G is such that χ(h) = ω(h), then G is a perfect graph. Perfect graphs have applications in many branches of mathematics and computer science, since they have a more precise structure than graphs in general. This structure can be used in proofs, as in Theorem 3.9. Theorem 3.8. Chordal graphs are perfect. Proof. Complete graphs are perfect, and by Theorem 3.7, we can construct any chordal graph by pasting two chordal graphs along a complete graph. Thus, we simply show that two perfect graphs pasted along a complete graph yields a perfect graph. Let G 1 and G 2 be perfect graphs and S be a complete graph contained in both G 1 and G 2. We show that G, which results from pasting G 1 and G 2 along S, is perfect. Let H be some induced subgraph of G. If we can show that χ(h) = ω(h), then G is perfect. By definition, χ(h) ω(h), so we simply show χ(h) ω(h). Let H 1 = G 1 H and H 2 = G 2 H, and let S = S H. Because H is an induced subgraph, all chords are included, so since S is complete, S is complete. Thus, H is made by pasting H 1 and H 2 along S. Because the H i are induced subgraphs of the G i, they are perfect, so they can be colored with ω(h i ) colors. Since S is complete, we relabel colors if necessary and can color H with max{ω(h 1 ), ω(h 2 )} colors. However, we see that m ω(h), so χ(h) ω(h), as desired. Thus, χ(h) = ω(h), so G is perfect.

30 20 Set Intersections, Perfect Graphs, and Voting in Agreeable Societies 3.6 (k, m)-agreeable Societies We now use the concept of a perfect graph to allow us to prove that the agreement number of a (k, m, n)-society is at least (k 1)n/(m 1). Theorem 3.9 (The (k, m)-agreeable Society Theorem). If G is the agreement graph of a (k, m, n)-society, then ( ) k 1 ω(g) n. m 1 In other words, in a (k, m)-agreeable society, there is some platform whose agreement proportion is at least (k 1)/(m 1). Note that this extends the Agreeable Society Theorem (in which k = 2, m = 3 and the guaranteed agreement proportion is 1/2) and the Super- Agreeable Society Theorem (in which k = m and the guaranteed agreeement proportion is 1). Proof. Because agreement graphs are perfect, χ(g) = ω(g), so we can color G using ω colors. Additionally, since G contains a K k, we know that ω(g) k. Thus, we can consider the k 1 largest color classes, which we ll call {C 1, C 2,..., C k 1 }. Let A = k 1 i=1 C i. The average order of a color class is n/ω, so the order of the union of the k 1 largest color classes is at least n(k 1)/ω. Thus, A n(k 1)/ω. Because A can be colored with only k 1 colors, no k members of A agree. Since this is a (k, m, n)- society, we know that A < m. Thus, m 1 n(k 1)/ω. Therefore, ω (k 1)n/(m 1). If we compare this bound to n/(m k + 1), the bound found in Theorem 3.5, we see that if k = 2 or k = m, then the bounds are the same. Otherwise, 2 < k < m, so (k 1)n/(m 1) is generally a better lower bound on the agreement number of a (k, m, n)-society. 3.7 Speculation and Open Questions As we have seen, set intersection theorems can provide a useful framework to model and understand the relationships between preference sets in many social contexts. Additionally, recent results in discrete geometry have social interpretations. The piercing number [7] of approval sets can be interpreted as the

31 Speculation and Open Questions 21 minimum number of platforms that are necessary such that everyone has some platform of which he or she approves. We suggest several directions which the reader may wish to explore. Thus far, we have considered approval voting on a single issue with infinitely many platforms. It is interesting to consider how to extend these conclusions to different situations. For example, what if we only had a finite number of platforms, rather than a continuous spectrum? Alternatively, we might wish to allow platforms to lie on a plane or a d-dimensional cube rather than a line, to more accurately represent multiple issues. Which of these theorems still apply, and which could be extended? A quick examination shows us that Theorem 3.3 works in d dimensions if we have a (d, d)-agreeable society (with convex approval sets) instead of a (2, 2)-agreeable society. This is simply due to Helly s theorem [14]. However, extending the other theorems to the d-dimensional context or the discrete setting may be more difficult. Additionally, we must examine our initial assumptions. While convexity seems to be a rational assumption in the linear case, in multiple dimensions, there are additional considerations that may need to be made. We have also assumed that voters place candidates along the spectrum in exactly the same manner, but people will not necessarily agree that one platform is more conservative than another. The original concept of an agreement graph could be applied to d-dimensional preferences, but we would like to be able to indicate that, for example, two people s preferences are the same in d 1 dimensions. To do so, one may wish to consider an agreement graph with weighted edges. Finally, we might wonder about the agreement parameters k and m for various issues which affect us personally. For instance, a society considering outlawing murder would probably be much more agreeable than that same society considering tax reform. Not only do the issues matter, however, but also the societies. Groups of similar people seem likely to be more agreeable than groups consisting of a more diverse population. Currently, we can empirically measure these parameters only by surveying large numbers of people about their preferences. It is interesting to speculate about methods for estimating k and m from limited data.

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33 Chapter 4 Convex Sets with k of Every m Meeting During the summer of 2005, Professor Su and I wrote another paper pertaining to this thesis, which looks at the same subject from a different angle. The paper is included here in its complete form. 4.1 Introduction In this article, we consider d-dimensional convex sets on a d-plane such that k out of every m intersect. We ask when we can guarantee that some fraction (say, a majority) of the sets will intersect and how many points {p i } are needed such that every set contains at least one point p i. Much of this work was inspired by the study of approval voting [3]. In a previous paper [3], we considered the d = 1 case. By Helly s theorem [14], for instance, if d = 1 and k = m = 2, then all sets have a nonempty intersection. By the (k, m)-agreeable Society Theorem [3], if there are n sets, then some ( k 1 m 1 ) n sets will have a non-empty intersection. In this paper, rather than considering the maximum number of sets that have a non-empty intersection, which is the agreement number a(s), we focus on the minimum number of points needed such that every set contains some point, called the piercing number Π(S). This concept was considered in depth in a series of papers by Alon and Kleitman, beginning with a paper from 1992 [2]. We develop relationships between these two numbers, the most notable being that a(s) + Π(S) n + 1 (Theorem 4.2) and a(s)π(s) n (Theorem 4.3).

34 24 Convex Sets with k of Every m Meeting 4.2 Definitions In this section, we fix some terminology and explain some of the basic concepts upon which our results rely. We begin by considering intersections of closed convex sets in dimension 1. In Figure 4.1, for ease of display, we have separated the sets {A i } vertically so that they can be distinguished. Figure 4.1: Linear Graph of a (2, 3, 6)-Society We define a (k, m, n)-society as a collection of n sets such that for any subset of m sets, there are at least k sets with a non-empty intersection. Notice that k m n. Unless otherwise specified, we assume k 2. If the order of the society does not matter, we simply call it a (k, m)-society. Due to the nature of (k, m)-societies, the following three statements are always true. These facts are easily verifiable and allow us to use the knowledge we have to generate bounds on societies for which we do not yet have information. 1. A (k, m)-society is a (k 1, m)-society 2. A (k, m)-society is a (k, m + 1)-society 3. A (k, m)-society is a (k 1, m 1)-society The agreement number of a point p in X, our set of platforms, is a(p), which is the number of sets which contain p. The agreement number a(s) of a society S is the maximum agreement number over all points in X, so a(s) = max p X a(p).

35 Definitions 25 Finally, the agreement set of S consists of points that are contained in a(s) sets. Figure 4.2 shows a society with agreement number 4, and the shaded rectangles cover the agreement set. Figure 4.2: Agreement Set of a Society with Agreement Number Four The piercing number of a collection C of sets, Π(C), is the minimum number of points necessary such that every set contains at least one point. A piercing set of C is a collection of points such that every set in C contains at least one point in the collection. If we wish to emphasize that C is d- dimensional, we write the piercing number as Π d (C). We sometimes need to know the largest possible piercing number given certain conditions, such as for a (k, m)-society. In this case, we write Π(k, m) or, to be more explicit, Π d (k, m). In certain cases, Π d (k, m) depends on n, the total number of sets; the remainder of the time, it depends solely on d, k, and m. Figure 4.3 depicts a collection of three sets such that each two intersect, but the intersection of all three sets is empty. This means that the piercing number must be at least two, and indeed, two points that pierce all sets are shown in the figure. The distance between sets, d(a, b), is the minimum number c such that there are sets v 1, v 2,..., v c for which the following intersections are all nonempty: A a A v1, A v1 A v2,..., A vc A b. In Figure 4.2, we see that the distance between sets 2 and 3, for example, is 1, because their intersection is empty, but sets 2 and 3 both have points in common with set 1. If no collection of sets will satisfy this condition, we say d(a, b) =. If d(a, b) = 0, then we say the sets are adjacent. In Figure 4.2, for instance, we see that sets 1 and 2 are adjacent. Upon closer inspection, we realize that set 1 is actually adjacent to every other set.

36 26 Convex Sets with k of Every m Meeting Figure 4.3: Collection of Sets with Piercing Number Two 4.3 Helly s Theorem A society is simply a set of sets, so work concerning set intersections can be applied to this problem. The most well known theorem in this area is Helly s theorem. This theorem was proven by Helly in 1913, but the result was not published until 1921, by Radon [14]. A proof of Helly s theorem may be found in [11]. Theorem 4.1 (Helly). Given t convex sets in R d where d < t, if every d + 1 of them intersect at a common point, then they all intersect at a common point. Besides Helly s theorem, another famous theorem about set intersections is the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma [10], which is concerned with set intersections on simplices. More recently, Niedermaier and Su [12] proved a set intersection theorem on trees that generalizes both Helly s theorem and the KKM lemma to this context. Since a line is a tree, their results are applicable to the d = 1 case. An excellent source of information on Helly s theorem and related theorems is Wenger s Helly-Type Theorems and Geometric Transversals [17]. This paper includes theorems, explanations, and open problems in the field. 4.4 Relationships Between Agreement and Piercing Numbers In this section, we introduce some theorems that relate the agreement numbers and piercing numbers of societies.

37 Piercing Numbers and Agreement Graphs 27 Theorem 4.2. The sum of a society s agreement number and its piercing number is no more than the order of the society plus one, so a(s) + Π(S) n + 1. Proof. Note that the agreement number of a society is equivalent to the maximum number of sets that can be pierced by a single point. Thus, if there are n sets in a society S, we can choose a point p in the intersection of a(s) sets and then one point for each of the n a(s) sets not containing p. Together, these are n a(s) + 1 points such that each set contains at least one of those points, so n a(s) + 1 Π(S). Theorem 4.3. The product of a society s agreement number and its piercing number is at least the order of the society, so a(s)π(s) n. Proof. We know a(s) represents the largest number of sets that can be pierced by a single point and Π(S) is the smallest number of points needed to ensure that each set contains at least one point. Therefore, there can be no more sets than a(s)π(s), which represents the largest number of sets in this configuration that could ever be pierced by Π(S) points. One nice property about the piercing number is that there are many inequalities which follow directly from the definition. For instance, Theorem 4.4. Π d (k, m) Π d 1 (k, m). Proof. Given a (k, m)-society in d 1 dimensions, we consider the (k, m)- society in d dimensions formed by taking the product of the sets of the original society and [0, 1]. This new society has the same piercing number as the original society. Since Π d (k, m) is the maximum possible piercing number of a (k, m)-society in d dimensions, Π d (k, m) Π d 1 (k, m). 4.5 Piercing Numbers and Agreement Graphs To further illustrate methods of finding the piercing number of a (k, m, n)- society, we consider the collection of sets shown in Figure 4.4. Figure 4.4 is a collection of 10 sets such that in any 3 sets, there is some point that is contained in some 2 of them. In other words, it is a (2, 3, 10)-society, which we call S. Theorem 4.5. The society in Figure 4.4 has piercing number 5.

38 28 Convex Sets with k of Every m Meeting Figure 4.4: A (2, 3, 10)-Society of Convex Sets Proof. Note that no point in S is contained in more than three convex sets. Thus, a(s) = 3. By Theorem 4.3, this means that Π 2 (S) n a(s) = 10 3 = 4. Suppose Π 2 (S) = 4. Since a(s) = 3 and there are 10 sets total, we see that at least two of the points must pierce three sets each, and that these sets must be unique. Therefore, these two points must be in the agreement set and must not be adjacent. Without loss of generality, we can choose the points shown in Figure 4.5. This leaves four sets which are not pierced and two points left to place. However, this configuration cannot be pierced with only two points, so Π 2 (S) > 4. We see that the five points where the large triangles meet pierce all of the sets, so Π 2 (S) = 5. If we are to understand other kinds of agreeability, it is helpful to examine methods of pictorially representing the agreeableness of a society. We construct an agreement graph G by letting each vertex represent a set and drawing an edge between two sets if the sets are adjacent. Notice that the distance between two sets (as defined earlier) is just the graph distance in the agreement graph. Moreover, if d(a, b) =, then there is no path from a to b, so we say that a and b are in different components of the agreement graph. Agreement graphs do not yield perfect information, as we see later, but we can determine whether the original graph is a (2, m)-society for any given m by examining the agreement graph. This is because agreement